Integrand size = 17, antiderivative size = 264 \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=\frac {a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}-\frac {3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac {3 a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {a^3 d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (\frac {a d}{b}+d x\right )}{2 b^6}+\frac {3 a^2 d \text {Chi}\left (\frac {a d}{b}+d x\right ) \sinh \left (c-\frac {a d}{b}\right )}{b^5}+\frac {\sinh (c+d x)}{b^3 d}+\frac {a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}+\frac {3 a^2 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^5}-\frac {3 a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{b^4}-\frac {a^3 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (\frac {a d}{b}+d x\right )}{2 b^6} \]
-3*a*Chi(a*d/b+d*x)*cosh(-c+a*d/b)/b^4-1/2*a^3*d^2*Chi(a*d/b+d*x)*cosh(-c+ a*d/b)/b^6+1/2*a^3*cosh(d*x+c)/b^4/(b*x+a)^2-3*a^2*cosh(d*x+c)/b^4/(b*x+a) +3*a^2*d*cosh(-c+a*d/b)*Shi(a*d/b+d*x)/b^5-3*a^2*d*Chi(a*d/b+d*x)*sinh(-c+ a*d/b)/b^5+3*a*Shi(a*d/b+d*x)*sinh(-c+a*d/b)/b^4+1/2*a^3*d^2*Shi(a*d/b+d*x )*sinh(-c+a*d/b)/b^6+sinh(d*x+c)/b^3/d+1/2*a^3*d*sinh(d*x+c)/b^5/(b*x+a)
Time = 0.63 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.89 \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {b \cosh (d x) \left (a^2 b d (5 a+6 b x) \cosh (c)-(a+b x) \left (2 a b^2+a^3 d^2+2 b^3 x\right ) \sinh (c)\right )-b \left ((a+b x) \left (2 a b^2+a^3 d^2+2 b^3 x\right ) \cosh (c)-a^2 b d (5 a+6 b x) \sinh (c)\right ) \sinh (d x)+a d (a+b x)^2 \left (\text {Chi}\left (d \left (\frac {a}{b}+x\right )\right ) \left (\left (6 b^2+a^2 d^2\right ) \cosh \left (c-\frac {a d}{b}\right )-6 a b d \sinh \left (c-\frac {a d}{b}\right )\right )+\left (-6 a b d \cosh \left (c-\frac {a d}{b}\right )+\left (6 b^2+a^2 d^2\right ) \sinh \left (c-\frac {a d}{b}\right )\right ) \text {Shi}\left (d \left (\frac {a}{b}+x\right )\right )\right )}{2 b^6 d (a+b x)^2} \]
-1/2*(b*Cosh[d*x]*(a^2*b*d*(5*a + 6*b*x)*Cosh[c] - (a + b*x)*(2*a*b^2 + a^ 3*d^2 + 2*b^3*x)*Sinh[c]) - b*((a + b*x)*(2*a*b^2 + a^3*d^2 + 2*b^3*x)*Cos h[c] - a^2*b*d*(5*a + 6*b*x)*Sinh[c])*Sinh[d*x] + a*d*(a + b*x)^2*(CoshInt egral[d*(a/b + x)]*((6*b^2 + a^2*d^2)*Cosh[c - (a*d)/b] - 6*a*b*d*Sinh[c - (a*d)/b]) + (-6*a*b*d*Cosh[c - (a*d)/b] + (6*b^2 + a^2*d^2)*Sinh[c - (a*d )/b])*SinhIntegral[d*(a/b + x)]))/(b^6*d*(a + b*x)^2)
Time = 0.89 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {a^3 \cosh (c+d x)}{b^3 (a+b x)^3}+\frac {3 a^2 \cosh (c+d x)}{b^3 (a+b x)^2}-\frac {3 a \cosh (c+d x)}{b^3 (a+b x)}+\frac {\cosh (c+d x)}{b^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 d^2 \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{2 b^6}-\frac {a^3 d^2 \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{2 b^6}+\frac {a^3 d \sinh (c+d x)}{2 b^5 (a+b x)}+\frac {a^3 \cosh (c+d x)}{2 b^4 (a+b x)^2}+\frac {3 a^2 d \sinh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^5}+\frac {3 a^2 d \cosh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^5}-\frac {3 a^2 \cosh (c+d x)}{b^4 (a+b x)}-\frac {3 a \cosh \left (c-\frac {a d}{b}\right ) \text {Chi}\left (x d+\frac {a d}{b}\right )}{b^4}-\frac {3 a \sinh \left (c-\frac {a d}{b}\right ) \text {Shi}\left (x d+\frac {a d}{b}\right )}{b^4}+\frac {\sinh (c+d x)}{b^3 d}\) |
(a^3*Cosh[c + d*x])/(2*b^4*(a + b*x)^2) - (3*a^2*Cosh[c + d*x])/(b^4*(a + b*x)) - (3*a*Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/b^4 - (a^3*d^2 *Cosh[c - (a*d)/b]*CoshIntegral[(a*d)/b + d*x])/(2*b^6) + (3*a^2*d*CoshInt egral[(a*d)/b + d*x]*Sinh[c - (a*d)/b])/b^5 + Sinh[c + d*x]/(b^3*d) + (a^3 *d*Sinh[c + d*x])/(2*b^5*(a + b*x)) + (3*a^2*d*Cosh[c - (a*d)/b]*SinhInteg ral[(a*d)/b + d*x])/b^5 - (3*a*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d* x])/b^4 - (a^3*d^2*Sinh[c - (a*d)/b]*SinhIntegral[(a*d)/b + d*x])/(2*b^6)
3.1.33.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(1054\) vs. \(2(262)=524\).
Time = 0.36 (sec) , antiderivative size = 1055, normalized size of antiderivative = 4.00
1/4*(2*exp(d*x+c)*b^5*x^2+2*exp(d*x+c)*a^2*b^3+exp(d*x+c)*a^3*b^2*d^2*x-6* exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^4*b*d^2-6*exp(d*x+c)*a^2*b^3* d*x+6*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^3*b^2*d-2*exp(-d*x-c)*a ^2*b^3-2*exp(-d*x-c)*b^5*x^2+exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^5* d^3-exp(-d*x-c)*a^4*b*d^2-5*exp(-d*x-c)*a^3*b^2*d-4*exp(-d*x-c)*a*b^4*x+6* exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a*b^4*d*x^2+12*exp((a*d-b*c)/b)*E i(1,d*x+c+(a*d-b*c)/b)*a^2*b^3*d*x+exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c) /b)*a^3*b^2*d^3*x^2+2*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^4*b*d^3 *x-6*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^2*b^3*d^2*x^2-12*exp(-(a *d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/b)*a^3*b^2*d^2*x+6*exp(-(a*d-b*c)/b)*Ei(1 ,-d*x-c-(a*d-b*c)/b)*a*b^4*d*x^2+12*exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c )/b)*a^2*b^3*d*x-exp(-d*x-c)*a^3*b^2*d^2*x+6*exp((a*d-b*c)/b)*Ei(1,d*x+c+( a*d-b*c)/b)*a^4*b*d^2-6*exp(-d*x-c)*a^2*b^3*d*x+6*exp((a*d-b*c)/b)*Ei(1,d* x+c+(a*d-b*c)/b)*a^3*b^2*d+exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^3*b^ 2*d^3*x^2+2*exp((a*d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^4*b*d^3*x+6*exp((a* d-b*c)/b)*Ei(1,d*x+c+(a*d-b*c)/b)*a^2*b^3*d^2*x^2+12*exp((a*d-b*c)/b)*Ei(1 ,d*x+c+(a*d-b*c)/b)*a^3*b^2*d^2*x+exp(-(a*d-b*c)/b)*Ei(1,-d*x-c-(a*d-b*c)/ b)*a^5*d^3+exp(d*x+c)*a^4*b*d^2-5*exp(d*x+c)*a^3*b^2*d+4*exp(d*x+c)*a*b^4* x)/d/b^6/(b*x+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 566 vs. \(2 (262) = 524\).
Time = 0.26 (sec) , antiderivative size = 566, normalized size of antiderivative = 2.14 \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {2 \, {\left (6 \, a^{2} b^{3} d x + 5 \, a^{3} b^{2} d\right )} \cosh \left (d x + c\right ) + {\left ({\left (a^{5} d^{3} - 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} - 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} - 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) + {\left (a^{5} d^{3} + 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} + 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} + 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \cosh \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (a^{4} b d^{2} + 2 \, b^{5} x^{2} + 2 \, a^{2} b^{3} + {\left (a^{3} b^{2} d^{2} + 4 \, a b^{4}\right )} x\right )} \sinh \left (d x + c\right ) - {\left ({\left (a^{5} d^{3} - 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} - 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} - 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) - {\left (a^{5} d^{3} + 6 \, a^{4} b d^{2} + 6 \, a^{3} b^{2} d + {\left (a^{3} b^{2} d^{3} + 6 \, a^{2} b^{3} d^{2} + 6 \, a b^{4} d\right )} x^{2} + 2 \, {\left (a^{4} b d^{3} + 6 \, a^{3} b^{2} d^{2} + 6 \, a^{2} b^{3} d\right )} x\right )} {\rm Ei}\left (-\frac {b d x + a d}{b}\right )\right )} \sinh \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (b^{8} d x^{2} + 2 \, a b^{7} d x + a^{2} b^{6} d\right )}} \]
-1/4*(2*(6*a^2*b^3*d*x + 5*a^3*b^2*d)*cosh(d*x + c) + ((a^5*d^3 - 6*a^4*b* d^2 + 6*a^3*b^2*d + (a^3*b^2*d^3 - 6*a^2*b^3*d^2 + 6*a*b^4*d)*x^2 + 2*(a^4 *b*d^3 - 6*a^3*b^2*d^2 + 6*a^2*b^3*d)*x)*Ei((b*d*x + a*d)/b) + (a^5*d^3 + 6*a^4*b*d^2 + 6*a^3*b^2*d + (a^3*b^2*d^3 + 6*a^2*b^3*d^2 + 6*a*b^4*d)*x^2 + 2*(a^4*b*d^3 + 6*a^3*b^2*d^2 + 6*a^2*b^3*d)*x)*Ei(-(b*d*x + a*d)/b))*cos h(-(b*c - a*d)/b) - 2*(a^4*b*d^2 + 2*b^5*x^2 + 2*a^2*b^3 + (a^3*b^2*d^2 + 4*a*b^4)*x)*sinh(d*x + c) - ((a^5*d^3 - 6*a^4*b*d^2 + 6*a^3*b^2*d + (a^3*b ^2*d^3 - 6*a^2*b^3*d^2 + 6*a*b^4*d)*x^2 + 2*(a^4*b*d^3 - 6*a^3*b^2*d^2 + 6 *a^2*b^3*d)*x)*Ei((b*d*x + a*d)/b) - (a^5*d^3 + 6*a^4*b*d^2 + 6*a^3*b^2*d + (a^3*b^2*d^3 + 6*a^2*b^3*d^2 + 6*a*b^4*d)*x^2 + 2*(a^4*b*d^3 + 6*a^3*b^2 *d^2 + 6*a^2*b^3*d)*x)*Ei(-(b*d*x + a*d)/b))*sinh(-(b*c - a*d)/b))/(b^8*d* x^2 + 2*a*b^7*d*x + a^2*b^6*d)
\[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=\int \frac {x^{3} \cosh {\left (c + d x \right )}}{\left (a + b x\right )^{3}}\, dx \]
\[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=\int { \frac {x^{3} \cosh \left (d x + c\right )}{{\left (b x + a\right )}^{3}} \,d x } \]
3/2*a^2*d*integrate(x*e^(d*x + c)/(b^5*d^2*x^4 + 4*a*b^4*d^2*x^3 + 6*a^2*b ^3*d^2*x^2 + 4*a^3*b^2*d^2*x + a^4*b*d^2), x) - 3/2*a^2*d*integrate(x/(b^5 *d^2*x^4*e^(d*x + c) + 4*a*b^4*d^2*x^3*e^(d*x + c) + 6*a^2*b^3*d^2*x^2*e^( d*x + c) + 4*a^3*b^2*d^2*x*e^(d*x + c) + a^4*b*d^2*e^(d*x + c)), x) - 3*a* b*integrate(x*e^(d*x + c)/(b^5*d^2*x^4 + 4*a*b^4*d^2*x^3 + 6*a^2*b^3*d^2*x ^2 + 4*a^3*b^2*d^2*x + a^4*b*d^2), x) - 3*a*b*integrate(x/(b^5*d^2*x^4*e^( d*x + c) + 4*a*b^4*d^2*x^3*e^(d*x + c) + 6*a^2*b^3*d^2*x^2*e^(d*x + c) + 4 *a^3*b^2*d^2*x*e^(d*x + c) + a^4*b*d^2*e^(d*x + c)), x) + 1/2*((b*d*x^3*e^ (2*c) - 3*a*x*e^(2*c))*e^(d*x) - (b*d*x^3 + 3*a*x)*e^(-d*x))/(b^4*d^2*x^3* e^c + 3*a*b^3*d^2*x^2*e^c + 3*a^2*b^2*d^2*x*e^c + a^3*b*d^2*e^c) - 3/2*a^2 *e^(-c + a*d/b)*exp_integral_e(4, (b*x + a)*d/b)/((b*x + a)^3*b^2*d^2) - 3 /2*a^2*e^(c - a*d/b)*exp_integral_e(4, -(b*x + a)*d/b)/((b*x + a)^3*b^2*d^ 2)
Leaf count of result is larger than twice the leaf count of optimal. 879 vs. \(2 (262) = 524\).
Time = 0.27 (sec) , antiderivative size = 879, normalized size of antiderivative = 3.33 \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=-\frac {a^{3} b^{2} d^{3} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{3} b^{2} d^{3} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 2 \, a^{4} b d^{3} x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 6 \, a^{2} b^{3} d^{2} x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 2 \, a^{4} b d^{3} x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 6 \, a^{2} b^{3} d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + a^{5} d^{3} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} - 12 \, a^{3} b^{2} d^{2} x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 6 \, a b^{4} d x^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + a^{5} d^{3} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 12 \, a^{3} b^{2} d^{2} x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 6 \, a b^{4} d x^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a^{3} b^{2} d^{2} x e^{\left (d x + c\right )} + a^{3} b^{2} d^{2} x e^{\left (-d x - c\right )} - 6 \, a^{4} b d^{2} {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 12 \, a^{2} b^{3} d x {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 6 \, a^{4} b d^{2} {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 12 \, a^{2} b^{3} d x {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} - a^{4} b d^{2} e^{\left (d x + c\right )} + 6 \, a^{2} b^{3} d x e^{\left (d x + c\right )} - 2 \, b^{5} x^{2} e^{\left (d x + c\right )} + a^{4} b d^{2} e^{\left (-d x - c\right )} + 6 \, a^{2} b^{3} d x e^{\left (-d x - c\right )} + 2 \, b^{5} x^{2} e^{\left (-d x - c\right )} + 6 \, a^{3} b^{2} d {\rm Ei}\left (\frac {b d x + a d}{b}\right ) e^{\left (c - \frac {a d}{b}\right )} + 6 \, a^{3} b^{2} d {\rm Ei}\left (-\frac {b d x + a d}{b}\right ) e^{\left (-c + \frac {a d}{b}\right )} + 5 \, a^{3} b^{2} d e^{\left (d x + c\right )} - 4 \, a b^{4} x e^{\left (d x + c\right )} + 5 \, a^{3} b^{2} d e^{\left (-d x - c\right )} + 4 \, a b^{4} x e^{\left (-d x - c\right )} - 2 \, a^{2} b^{3} e^{\left (d x + c\right )} + 2 \, a^{2} b^{3} e^{\left (-d x - c\right )}}{4 \, {\left (b^{8} d x^{2} + 2 \, a b^{7} d x + a^{2} b^{6} d\right )}} \]
-1/4*(a^3*b^2*d^3*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + a^3*b^2*d^3*x^2* Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 2*a^4*b*d^3*x*Ei((b*d*x + a*d)/b)*e^ (c - a*d/b) - 6*a^2*b^3*d^2*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 2*a^4* b*d^3*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 6*a^2*b^3*d^2*x^2*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + a^5*d^3*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) - 12 *a^3*b^2*d^2*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 6*a*b^4*d*x^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + a^5*d^3*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 1 2*a^3*b^2*d^2*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 6*a*b^4*d*x^2*Ei(-(b *d*x + a*d)/b)*e^(-c + a*d/b) - a^3*b^2*d^2*x*e^(d*x + c) + a^3*b^2*d^2*x* e^(-d*x - c) - 6*a^4*b*d^2*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 12*a^2*b^3* d*x*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 6*a^4*b*d^2*Ei(-(b*d*x + a*d)/b)*e ^(-c + a*d/b) + 12*a^2*b^3*d*x*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) - a^4*b *d^2*e^(d*x + c) + 6*a^2*b^3*d*x*e^(d*x + c) - 2*b^5*x^2*e^(d*x + c) + a^4 *b*d^2*e^(-d*x - c) + 6*a^2*b^3*d*x*e^(-d*x - c) + 2*b^5*x^2*e^(-d*x - c) + 6*a^3*b^2*d*Ei((b*d*x + a*d)/b)*e^(c - a*d/b) + 6*a^3*b^2*d*Ei(-(b*d*x + a*d)/b)*e^(-c + a*d/b) + 5*a^3*b^2*d*e^(d*x + c) - 4*a*b^4*x*e^(d*x + c) + 5*a^3*b^2*d*e^(-d*x - c) + 4*a*b^4*x*e^(-d*x - c) - 2*a^2*b^3*e^(d*x + c ) + 2*a^2*b^3*e^(-d*x - c))/(b^8*d*x^2 + 2*a*b^7*d*x + a^2*b^6*d)
Timed out. \[ \int \frac {x^3 \cosh (c+d x)}{(a+b x)^3} \, dx=\int \frac {x^3\,\mathrm {cosh}\left (c+d\,x\right )}{{\left (a+b\,x\right )}^3} \,d x \]